EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES

Commun. Korean Math. Soc. (, No., pp. 1 https://doi.org/1.4134/ckms.c1739 pissn: 1225-1763 / eissn: 2234-324 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES Muhammad Arshad, Shahid Mubeen, Kottakkaran Sooppy Nisar, and Gauhar Rahman Abstract. In this present paper, our aim is to introduce an extended Wright-Bessel function Jα,q λ,γ,c (z which is established with the help of the extended beta function. Also, we investigate certain integral transforms and generalized integration formulas for the newly defined extended Wright-Bessel function Jα,q λ,γ,c (z and the obtained results are expressed in terms of Fox-Wright function. Some interesting special cases involving an extended Mittag-Leffler functions are deduced. 1. Introduction The well known Wright Bessel function (or Bessel-Maitland function, misnamed after the second name Maitland of E. M. Wright defined in [24] as: Jα(z λ ( z n (1.1 ; λ > 1, z C. Γ(α + λn + 1n! n Pathak [19] gave the generalization of (1.1 as: Jα,β(z λ ( 1 n ( z (1.2 2 α+2β+2n Γ(β + n + 1Γ(α + β + λn + 1, n where λ >, z C \ (,, α, β C. Singh et al. [22] defined another generalization of Wright-Bessel function in the following form: Jα,q λ,γ (γ qn ( z n (1.3 (z Γ(α + λn + 1n!, n where α, λ, γ C, R(λ >, R(α > 1, R(γ > and q (, 1 N and (γ 1 and (γ nq Γ(γ+nq Γ(γ is the generalized Pochhammer symbol (see [13]. Received February 8, 217; Accepted March 14, 217. 21 Mathematics Subject Classification. 33B2, 33C2, 33C45, 33C6, 33B15, 26A33. Key words and phrases. gamma function, generalized hypergeometric function, Oberhettinger formula, Wright-Bessel function, generalized Wright function, Mittag-Leffler function. 1 c Korean Mathematical Society

2 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Clearly, if q, then (1.3 reduces to (1.1. We recall the generalized hypergeometric function p F q (z is defined in [5] as: pf q (z p F q, (α 1, (α 2, (α p ; z (1.4 (β 1, (β 2, (β q (α 1 n (α 2 n (α p n z n (β 1 n (β 2 n (β q n n!, n where α i, β j C; i 1, 2,..., p, j 1, 2,..., q and b j, 1, 2,... and (z n is the Pochhammer symbols. The gamma function is defined as: (1.5 Γ(µ t µ 1 e t dt, µ C, (1.6 Γ(z + 1 zγ(z, z C, and beta function is defined as: (1.7 B(x, y t x 1 (1 t y 1 dt. For i 1, 2,..., p and j 1, 2,..., q, the Wright type hypergeometric function is defined (see [24]-[26] by the following series as: (α i, A i 1,p pψ q (z p Ψ q ; z (1.8 (β j, B j 1,q Γ(α 1 + A 1 n Γ(α p + A p n z n Γ(β 1 + B 1 n Γ(β q + B q n n!, n where β r and µ s are real positive numbers such that q p 1 + β s α r >. s1 r1 Equation (4.1 differs from the generalized hypergeometric function p F q (z defined (3.3 only by a constant multiplier. The generalized hypergeometric function p F q (z is a special case of p Ψ q (z for A i B j 1, where i 1, 2,..., p and j 1, 2,..., q: (α 1 1,... (α p (α (1.9 q pf q ; z 1 i, 1 1,p p pψ q ; z. Γ(β j (β 1,... (β q Γ(α i (β j, 1 1,q j1 Recently Sharma and Devi [21] defined extended Wright type function in the following series: (α i, A i 1,p, (γ, 1 p+1ψ q+1 (z, p p+1 Ψ q+1 ; (z, p (β j, B j 1,q, (c, 1 i1

(1.1 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 3 Γ(α 1 + A 1 n Γ(α p + A p n B p (γ + n, c γz n, Γ(β 1 + B 1 n Γ(β q + B q n Γ(c γn! n where i 1, 2,..., p and j 1, 2,..., q, R(p >, R(c > R(γ > and p. In this paper, we introduce an extended Wright-Bessel (Bessel-Maitland function as defined in (1.3 as: Jα,q λ,γ,c B p (γ + nq, c γ (c nq ( z n (1.11 (z, p B(γ, c γ Γ(α + λn + 1n!, n where p, α, λ, γ, c C, R(p >, R(λ, R(α > 1, R(γ >, R(c > and B p (x, y is an extended beta function (see [3], [2]. Remark 1.1. i If we set p in (1.11, then extended Wright-Bessel function reduces to the (1.3. ii If we set p, q in (1.11, then extended Wright-Bessel function reduces to the (1.1. iii If α is replaced by α 1 and z by z, then (1.11 reduces to J λ,γ,c α 1,q ( z Eγ,q,c (z which is an extended Mittag-Leffler function (see [14]. iv If q 1 and α is replaced by α 1 and z by z, then (1.11 reduces to J λ,γ,c α 1,1 ( z Eγ,c (z which is an extended Mittag-Leffler function (see [18]. For this continuation of our study, we recall the following result of Oberhettinger [17] (1.12 ( z α 1 z + b + β z 2 + 2bz dz ( α b 2βb β Γ(2αΓ(β α, < R(α < R(β. 2 Γ(1 + α + β For various other investigation containing special function, the reader may refer to the recent work of researchers (see [1], [4], [6], [12], [15], [16]. In the investigation of various properties and relations of the function Jα,q λ,γ,c (z as defined in (1.11, we need the following well known facts. Sneddon [23] defined Beta Euler and Laplace transforms in following forms: (1.13 B(f(z; α, β where R(α > and R(β >. (1.14 respectively. L{f(z} z α 1 (1 z β 1 f(zdz, e sz f(zdz,

4 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Meijer [11] defined the following K-transform integral equation as: (1.15 R v {f(x; p} (px 1 2 Kv (pxf(xdx, where the parameter p C and K v (z represent a modified Bessel function of third kind defined in [11]. Varma transform (see [2] is defined by the following integral equation as: (1.16 V (f, k, m; s (sx m 1 2 Wk,m (sxf(xdx, where W k,m (z represent Wittaker function (see [9, p. 55]. 2. Integral transforms of an extended Bessel function In this section, we investigate some integral transforms such as Beta, Laplace, Varma and K-transforms of an extended Wright-Bessel function as defined in (1.11 and we express the obtained result in terms of an extended Wright-type hypergeometric function (1.1. Theorem 2.1. Let α, λ, γ, c, µ, δ, ν C, R(α 1, R(λ, R(γ, R(µ, R(c, R(ν, p, q (, 1 \ N, then the following integral transforms holds: (2.1 z µ 1 (1 z ν 1 Jα,q λ,γ,c (xz δ, pdz Γ(ν (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α + 1, λ, (µ + ν, δ, (c, 1 ; Proof. From (1.11 and (1.13, we obtain ( x, p z µ 1 (1 z ν 1 Jα,q λ,γ,c (xz δ, pdz ( z µ 1 (1 z ν 1 B p (γ + nq, c γ (c nq ( 1 n (xz δ n dz. B(γ, c γ Γ(α + λn + 1n! n Interchanging the order of integration and summation, we have n z µ 1 (1 z ν 1 J λ,γ,c (xz δ, pdz α,q B p (γ + nq, c γ (c nq ( 1 n (x n z µ+δn 1 (1 z ν 1 dz B(γ, c γ Γ(α + λn + 1n! n B p (γ + nq, c γ (c nq ( 1 n (x n Γ(µ + δγ(ν B(γ, c γ Γ(α + λn + 1n! Γ(µ + ν + δ.

EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 5 Γ(ν B p (γ + nq, c γ Γ(c + nq( 1 n (x n Γ(µ + δ Γ(γ Γ(c γ Γ(α + λn + 1n! Γ(µ + ν + δ, n which upon using (1.1, we get the required result. From above theorem, we observe the following two cases. Case (i. If λ δ and µ α + 1, then (2.1 reduces to the following result: (2.2 z α (1 z ν 1 J λ,γ,c (xz λ, pdz Γ(νJ δ,γ,c α,q α+ν,q(x and Case (ii. If λ δ and µ α + 1 and z (1 z, then (2.1 reduces to the following result: (2.3 z µ 1 (1 z α J λ,γ,c (x(1 z λ, pdz Γ(µJ δ,γ,c α,q α+µ,q(x. Theorem 2.2. Let α, λ, γ, c, µ, δ C, R(α 1, R(λ, R(γ, R(µ, R(c, R(s, p, q (, 1 \ N, then the following integral transforms holds: (2.4 z µ 1 e sz Jα,q λ,γ,c (xz δ, pdz s µ Γ(γ Proof. From (1.11 and (1.14, we have z µ 1 e sz J λ,γ,c (xz δ, pdz α,q z µ 1 e sz ( n 3Ψ 2 (c, q, (µ, δ, (γ, 1 ; ( x, p s. δ (α + 1, λ, (c, 1 ; B p(γ+nq,c γ (c nq( 1 n (xz δ n B(γ,c γ Γ(α+λn+1n! Interchanging the order of integration and summation, we have z µ 1 e sz J λ,γ,c (xz δ, pdz α,q dz. B p (γ + nq, c γ (c nq ( 1 n (x n z µ+δn 1 e sz dz B(γ, c γ Γ(α + λn + 1n! n B p (γ + nq, c γ (c nq ( 1 n (x n B(γ, c γ Γ(α + λn + 1n! s µ δn Γ(µ + δn n s µ B p (γ + nq, c γ Γ(c + nq( 1 n (x n Γ(µ + δn, Γ(γ Γ(c γ Γ(α + λn + 1n! n which by using (1.1, we get the required result. From above theorem, we observe the following two particular cases:

6 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN (2.5 (i If p, q 1, µ α + 1, λ δ, then (2.4 reduces to by using the relation z α e sz J λ,γ α,1 (xzλ dz s α 1 (1 + xs λ γ, n (a nz n n! (1 z a (see [2], and (ii If p, γ q 1, µ α + 1, λ δ and x ±t, then (2.4 reduces to ( n z α e sz J λ,1 t α,1 ( zλ dz s α 1 (2.6 n s α 1 1 t ; s λ sλ α 1 s λ ± t. s λ t < 1 Theorem 2.3. Let α, λ, δ, ρ, γ, c C and R(α 1, R(γ, R(δ, R(ρ, R(c, p and q N, then the following result holds: (2.7 t δ 1 K µ (stjα,q λ,γ,c (ωt ρ, pdt (c, q, (δ ± µ, ρ, (γ, 1 ; s δ Γ(γ 4 Ψ 2 (α + 1, λ, (c, 1 ; 2δ 2 Proof. From (1.11 and (1.15, we have t δ 1 K µ (stjα,q λ,γ,c (ωt ρ dt ( t δ 1 K µ (st s λ ( ω( 2 s ρ, p B p(γ+nq,c γ (c nq( ωt ρ n B(γ,c γ Γ(α+λn+1n! n. dt. By changing the variable t z s and interchanging of sum and integration, we have ( ( z B p (γ + nq, c γ (c nq ( ω n ( z s δ 1 s n 1 K µ (z ρ B(γ, c γ Γ(α + λn + 1n! s dz n s δ B p (γ + nq, c γ (c nq ( ω s n ρ B(γ, c γ Γ(α + λn + 1n! n ( z s δ+ρn 1 K µ (zdz. Now, by using the following Mathai and Saxena ([1, p. 78] formula (2.8 x ρ 1 K µ (xdx 2 ρ 2 Γ( ρ ± µ, 2 in the above equation, we obtain ( ( z B p (γ + nq, c γ (c nq ( ω n ( z s δ 1 s n 1 K µ (z ρ B(γ, c γ Γ(α + λn + 1n! s dz n

EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 7 1 B p (γ + nq, c γ Γ(c + nq( ω Γ(γ s δ s n ρ Γ(c γ Γ(α + λn + 1n! 2δ ρn 2 Γ(δ + ρn ± µ n which by using (1.1, we get the required result. Theorem 2.4. Let α, λ, δ, ρ, γ, µ, νc C and R(α 1, R(γ, R(δ, R(ρ, R(c, p and q N, then the following result holds: (2.9 e st 2 t δ 1 W µ,ν (stjα,q λ,γ,c (ωt ρ, pdt (c, q, ( 1 1 s δ Γ(γ. 2 4Ψ 3 ± ν + δ, ρ, (γ, 1 ; ( ω s, p. ρ (α + 1, λ, (1 µ + δ, ρ, (c, 1 ; Proof. From (1.11 and (1.16, we have e st e st 2 t δ 1 W µ,ν (stjα,q λ,γ,c (ωt ρ dt ( 2 t δ 1 B p (γ + nq, c γ (c nq ( ωt ρ n W µ,ν (st dt. B(γ, c γ Γ(α + λn + 1n! n By changing the variable st z and interchanging of sum and integration in above equation yields e z z 2 ( s δ 1 W µ,ν (zjα,q λ,γ,c (t ρ 1 s dz s δ B p (γ + nq, c γ (c nq ( ω s n ( ρ e z z 2 ( B(γ, c γ Γ(α + λn + 1n! s δ+ρn 1 W µ,ν (zdz. n Now, by using the formula (2.1 e s 2 s β 1 W µ,ν (sds Γ( 1 2 + ν + βγ( 1 2 ν + β, Γ(1 µ + β (where R(β ± ν > 1 2 in above equation (see Mathai and Saxena [1, p. 79], we get e z z 2 ( s δ 1 W µ,ν (zjα,q λ,γ,c (t ρ 1 s dz B p (γ + nq, c γ Γ(c + nq( ω s n Γ( 1 ρ 2 ± ν + δ + ρn Γ(γ Γ(c γ Γ(α + λn + 1n! Γ(1 µ + δ + ρn s δ n which by using (1.1, we get the required result.

8 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN 3. Unified integral formulas of an extended Bessel function In this section, we establish two generalized integral formulas containing extended Wright-Bessel function as defined in (1.11 which is represented in terms of Wright-type function defined in (1.1 by inserting with the suitable argument defined in (1.12. Theorem 3.1. Let α, µ, δ, v, λ, γ, c C with R(δ >, R(α > 1, R(λ >, R(γ >, R(c >, R(µ > R(δ >, p, q (, 1 N and z >, then the following result holds: ( z δ 1 z + b + ( µ z 2 + 2bz J λ,γ,c y α,q z + b + dz z 2 + 2bz Γ(2δbδ µ (µ + 1; 1, (µ δ, 1, (c, q, (γ, 1; (3.1 2 δ 1 Γ(γ 4 Ψ 4 ( y b, p. (α + 1, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 Proof. Let L 1 be the left hand side of Theorem (3.1 and applying (1.11 to the integrand of (3.1, we have L 1 n z δ 1 ( z + b + z 2 + 2bz µ B p (γ + nq, c γ(c nq ( y B(γ, c γγ(λn + α + 1n! (z+b+ z 2 +2bz n dz. By interchanging the order of integration and summation, which is verified by the uniform convergence of the series under the given assumption of Theorem 3.1, we have (3.2 L 1 B p (γ + nq, c γ(c nq ( 1 n y n ( z δ 1 z + b + (µ+n z 2 + 2bz dz. B(γ, c γγ(λn + α + 1n! n By considering the assumption given in theorem 3.1, since R(µ+n > R(δ >, R(α > 1 and applying (1.12 to (3.2, we obtain L 1 B p (γ + nq, c γ(c nq ( 1 n y n 2(µ + nb (µ+n ( b Γ(2δΓ(µ + n α δ B(γ, c γγ(λn + α + 1n! 2 Γ(1 + µ + δ + n. n Applying (1.6, we get L 1 Γ(2δbδ µ ( 1 n y n B p (γ + nq, c γγ(c + nqγ(µ + n + 1Γ(µ + n δ 2 δ 1 Γ(γ Γ(c γn! b n Γ(α + λn + 1Γ(µ + nγ(δ + µ + n + 1 n which upon using (1.1, we get the required result. Theorem 3.2. Let α, µ, δ, v, λ, γ, c C with R(δ >, R(α > 1, R(λ >, R(γ >, R(c >, R(µ > R(δ >, p, q (, 1 N and z >. Then

EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 9 the following result holds: z δ 1 ( z + b + z 2 + 2bz ( µ J λ,γ,c yz α,q z+b+ z 2 +2bz dz (µ + 1; 1, (2δ, 2, (c, q, (γ, 1; Γ(µ δbδ µ (3.3 2 δ 1 4Ψ 4 ( y 2 Γ(γ, p. (µ, 1, (α + 1, λ, (µ + δ + 1, 2, (c, 1 Proof. Let L 2 be the left hand side of (3.2 and applying (1.11 to the integrand of (3.3, we have L 2 n z δ 1 ( z + b + z 2 + 2bz µ B p (γ + nq, c γ(c ( nq B(γ, c γγ(α + λn + 1n! yz z+b+ z 2 +2bz n dz. By interchanging the order of integration and summation, which is verified by the uniform convergence of the series under the given assumption of Theorem 3.2, we have B p (γ + nq, c γ(c nq ( 1 n y n L 2 B(γ, c γγ(α + λn + 1n! (3.4 n z δ+n 1 ( z + b + z 2 + 2bz (µ+n dz. By considering the assumption given in Theorem 3.2, since R(β > R(α >, R(α > 1 and applying (1.12 to (3.4, we obtain L 2 B p (γ + nq, c γ(c nq ( 1 n y n 2(µ + nb δ µ 1 + 2nΓ(µ δ ( Γ(2δ B(γ, c γγ(α + λn + 1n! 2δ+n Γ(1 + µ + δ + 2n n Γ(µ δbδ µ B p (γ + nq, c γ( 1 n y n Γ(c + nqγ(2δ + 2nΓ(µ + n + 1 2 δ 1 Γ(γ 2 n Γ(c γ Γ(α + λn + 1Γ(µ + nγ(1 + µ + δ + 2n n which upon using (1.1, we get the required result. 4. Special cases In this section, we present the extended form of generalized Mittag-Leffler functions which are the special cases of extended Wright-Bessel function defined in (1.11. Also, we prove four corollaries which are the special cases of obtained Theorems 3.1 and 3.2. Case 1. If α is replaced by α 1 and z by z, then (1.11 reduces to an extended Mittag-Leffler function defined as: (4.1 J λ,γ,c α 1,q ( z Eγ,q,c (z B p (γ + nq, c γ(c nq B(γ, c γγ(λn + αn!. n

1 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Case 2. If q 1 and α is replaced by α 1 and z by z, then (1.11 reduces to an extended Mittag-Leffler function (4.2 J λ,γ,c α 1,1 ( z Eγ,c (z B p (γ + n, c γ(c n B(γ, c γγ(λn + αn!. n Corollary 4.1. Let the conditions of Theorem 2.1, then the following integral transforms holds: (4.3 z µ 1 (1 z ν 1 E γ,q,c (xzδ, pdz Γ(ν (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α, λ, (µ + ν, δ, (c, 1 ; (x, p Corollary 4.2. Let the conditions of Theorem 2.1, then the following integral transforms holds: (4.4 z µ 1 (1 z ν 1 E γ,c (xzδ, pdz Γ(ν (c, 1, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α, λ, (µ + ν, δ, (c, 1 ; (x, p Corollary 4.3. Let the conditions of Theorem 2.2 are satisfied, then the following integral transforms holds: (4.5 z µ 1 e sz E γ,q,c (xzδ, pdz (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 2 (α, λ, (c, 1 ; s µ ( x s δ, p Corollary 4.4. Let the conditions of Theorem 2.2 are satisfied, then the following integral transforms holds: (4.6 z µ 1 e sz E γ,c (xzδ, pdz (c, 1, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 2 (α, λ, (c, 1 ; s µ ( x s δ, p Similarly, we can derives some corollaries by using the conditions of Theorems 2.3 and 2.4. Corollary 4.5. Assume that the conditions of Theorem 3.1 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,q,c y z + b + dz z 2 + 2bz....

(4.7 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 11 (µ + 1; 1, (µ δ, 1, (c, q, (γ, 1; Γ(2δbδ µ 2 δ 1 Γ(γ 4 Ψ 4 (α, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 ( y b, p Corollary 4.6. Assume that the conditions of Theorem 3.1 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,c y z + b + dz z 2 + 2bz (µ + 1; 1, (µ δ, 1, (c, 1, (γ, 1; (4.8 Γ(2δbδ µ 2 δ 1 Γ(γ 4 Ψ 4 ( y b, p. (α, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 Corollary 4.7. Assume that the conditions of Theorem 3.2 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,q,c yz z + b + dz z 2 + 2bz Γ(µ δbδ µ 2 δ 1 4Ψ 4 (µ + 1; 1, (2δ, 2, (c, q, (γ, 1; (4.9 ( y 2 Γ(γ, p. (µ, 1, (α, λ, (µ + δ + 1, 2, (c, 1 Corollary 4.8. Assume that the conditions of Theorem 3.2 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,c yz z + b + dz z 2 + 2bz Γ(µ (µ + 1; 1, (2δ, 2, (c, 1, (γ, 1; δbδ µ (4.1 2 δ 1 4Ψ 4 ( y 2 Γ(γ, p. (µ, 1, (α, λ, (µ + δ + 1, 2, (c, 1 Concluding Remark. In this paper, we introduced certain integral transforms and representation formulas for newly defined extended Wright-Bessel function and express the obtained results in terms of an extended Wright-type hypergeometric function. If letting p, then we have the results of the generalized Wright-Bessel function and Mittag-Leffler functions proved in ([7], [22]. References [1] M. S. Abouzaid, A. H. Abusufian, and K. S. Nisar, Some unified integrals associated with generalized Bessel-Maitland function, Int. Bull. Math. Research 3 (216, 18 23. [2] M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (24, no. 2, 589 62. [3] M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press, Boca Raton, 22. [4] J. Choi, P. Agarwal, S. Mathur, and S. D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc. 51 (214, no. 4, 995 13. [5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, NewYork, Toronto, London, 1953..

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EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 13 Shahid Mubeen Department of Mathematics University of Sargodha Sargodha, Pakistan E-mail address: smjhanda@gmail.com Kottakkaran Sooppy Nisar Department of Mathematics College of Arts and Science-Wadi Aldawaser 11991, Prince Sattam bin Abdulaziz University Alkharj, Kingdom of Saudi Arabia E-mail address: ksnisar1@gmail.com, n.sooppy@psau.edu.sa Gauhar Rahman Department of Mathematics International Islamic University Islamabad, Pakistan E-mail address: gauhar55uom@gmail.com